Problem: Simplify and expand the following expression: $ \dfrac{q + 1}{4q - 3}+\dfrac{5q}{3q + 9} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4q - 3)(3q + 9)$ Multiply the first term by $\dfrac{3q + 9}{3q + 9}$ $ \begin{align*} \dfrac{q + 1}{4q - 3} \times \dfrac{3q + 9}{3q + 9} & = \dfrac{(q + 1)(3q + 9)}{(4q - 3)(3q + 9)} \\ & = \dfrac{3q^2 + 12q + 9}{(4q - 3)(3q + 9)}\end{align*} $ Multiply the second term by $\dfrac{4q - 3}{4q - 3}$ $ \begin{align*} \dfrac{5q}{3q + 9} \times \dfrac{4q - 3}{4q - 3} & = \dfrac{(5q)(4q - 3)}{(3q + 9)(4q - 3)} \\ & = \dfrac{20q^2 - 15q}{(3q + 9)(4q - 3)}\end{align*} $ Now we have: $ = \dfrac{3q^2 + 12q + 9}{(4q - 3)(3q + 9)} + \dfrac{20q^2 - 15q}{(3q + 9)(4q - 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3q^2 + 12q + 9 + 20q^2 - 15q}{(4q - 3)(3q + 9)} $ $ = \dfrac{23q^2 - 3q + 9}{(4q - 3)(3q + 9)}$ Expand the denominator: $ = \dfrac{23q^2 - 3q + 9}{12q^2 + 27q - 27}$